129
HUMPHREY,
J. H. & LIGHTBOWN,
J. W. (1952). J . gen. Microbiol. 7 , 129-143
A General Theory for Plate Assay of Antibiotics with
some Practical Applications
BY J. H. HUMPHREY
AND
J. W. LIGHTBOWN
National Institute for Medical Research, Mill Hill, London, N . W . 7
SUMMARY: The distribution of antibiotic (or other substance) in the agar around
a container or around a hole in a punch-plate can be expressed theoretically by an
equation involving: the initial quantity of antibiotic, the depth of the agar layer,
the diffusion constant, the concentration a t a given distance from the container, and
the time of diffusion. The validity of the equation was confirmed by measurement
of the diffusion constants of penicillin, streptomycin and aureomycin, and of the
critical concentration of these substances required t o inhibit test organisms, followed
by the use of the values so obtained t o predict the sizes of the inhibition zones
produced experimentally by these antibiotics after varying periods of diffusion.
The theory predicts that the square of the inhibition zone diameter will be
proportional t o the logarithm of the antibiotic concentration. This relationship was
found t o hold, when accurate assays were made, for a number of antibiotics but not
for penicillin when tested with Bacillus subtilis. The most important factor determining
the slope of the dose-response curve under given conditions is the diffusion constant
of the antibiotic. The slope can, however, be increased by prolonging the time allowed
for diffiusion.
Particular factors which affected both the sharpness of the zone edge and the
nature of the dose-response curve were production of small amounts of penicillinase by
strains of B . subtilis used for penicillin assay, and uptake of streptomycin by
organisms used for streptomycin assay. Measurements of adsorption of streptomycin by B. subtilis, B. pumilus and by Staphylococcus aureus were made, and were
shown t o fit equations of the Freundlich isotherm type.
Although very good plate assay methods are available for penicillin and
streptomycin, those described for the newer antibiotics are less satisfactory
owing to a tendency for the zone edges to be poorly defined and to shift their
position as incubation of the plates is continued. Unless a valid theoretical
treatment of the nature of diffusion of an antibiotic from the cup into the
surrounding agar gel is available, attempts to investigate the factors governing
the nature of the zone edge and the slope of the dose-response curve for new
antibiotics will be empirical and time-consuming. The only theoretical treatment of this problem of which we were aware was that given by Cooper &
Woodman (1946), who discussed diffusion of antiseptics and of penicillin from
a Heatley cup into agar, and claimed good agreement between predicted zone
diameters and those which they observed in practice. They used, however,
a formula derived from considerations of linear, not radial, diffusion, and
assumed that the concentration of antibiotic in the cup was constant during
the first 8 hr. These assumptions do not hold in plate assays as often performed,
and we decided therefore to investigate the theoretical treatment from first
principles. With the aid of certain assumptions, which are discussed below,
a theoretical expression was derived, differing somewhat from that given by
9
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J . H . Humphrey and J . W. Lightbown
Cooper & Woodman; its validity was tested in various ways. These tests
revealed that in penicillin and streptomycin assays, under certain conditions,
there is a sudden fall in concentration of antibiotic a t the zone boundary,
which results in a very sharp edge. Tests were also made to discover the reason
for this boundary effect.
METHODS
Fish spine beads, size no. 2 (obtainable from Messrs Taylor, Tunnicliffe and
Co., 125 High Holborn, London, W.C. 1) were used (Stewart & Thorpe (to be
published)). These beads are very uniform, and may be assumed to hold on
the average 0.025 ml.
Assay trays. Large flat trays, 10 x 10 in. accommodating 64 beads were used
(Kantorowicz, 1951).
Organisms. Spores of B. subtilis, I.C.I. strain (NCTC 8236) and B. pumilus
Mill Hill strain (NCTC 8241) were used for most assays. Suspensions were
incorporated into molten agar at 60-70°, at a density of approximately
3 x lo7 and 6 x lo7 viable spores/ml. of medium respectively.
For certain experiments suspensions of Staphylococcus aureus (NCTC 7447)
or of Sarcina Zutea (NCTC 8340) were used. Suspensions were made with
a density equivalent to Brown's tube no. 6 (Burroughs Wellcome Ltd.) and to
ten times this strength respectively, and were incorporated into molten agar
at 45' a t a concentration of 1 yo (v/v).
The strains used had been subjected to several single colony isolations
and were stored freeze-dried, except for the B. subtilis and B. pumilus strains
which were stored at 2' as spore suspensions.
Media. Experiments with penicillin and aureomycin were performed with
a Lemco Marmite peptone agar p H 7 when B. subtiEis I.C.I. strain was used,
and with the enriched seed layer medium prescribed by the Food and Drugs
Administration of the U.S. Federal Security Agency (1951) when other
organisms were used.
Experiments with streptomycin and dihydrostreptomycin were carried out
in nutrient agar containing peptone, beef extract and yeast extract pH7.8.
Difco agar 2 yo was used in all media.
Zone diameters were measured on a screen after x 10 magnification. Under
the conditions of our assays the coefficient of variation of the diameter of
a single zone was repeatedly found to be approximately 1 yo.
Diffusion constants. These were measured by the method of Friedman &
Kraemer (1930). I n this method a layer of agar gel is formed in a cylindrical
container, and a n equal volume of liquid is placed above it. The liquid layer
is stirred continuously. Initially the substance whose diffusion constant is to be
measured is either wholly in the agar layer or wholly in the liquid layer. By
measurement of the change in concentration in the liquid layer, and the use
of a theoretical expression derived by March & Weaver (1928) for this system,
the diffusion constant can be obtained. It should be noted that the theoretical
expression quoted by Friedman & Kraemer is wrong (although the curve
which they give is correct), and the proper equation relating z), the fraction
'
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Theory of plate assays
131
diffused across the boundary, and the difiusion constant k is as follows (taken
from March & Weaver) :
v = - C(n2Bi/4Zi2)e-zi*T,
(1)
where T = kt/a2. I n this expression a =depth of agar layer, k =diffusion constant, t = time, 2 is given by the roots of tail Z+Z=O. B is a function whose
values are B, = 0.545, Bl = 0.748, U , = 0-789, B, = 0.799, B4 and higher
terms = 0.810 ( = 8 / n 2 ) .
This equation, expanded to 10 ternis, bec*onies
V=L-
[0.32’7e-4.117T + 0.0766e-24.14T + 0.0306 e-63’68r + 0.016e-123r + 0.01 e-2ooT’
+0~0067e-300T+0~0048e-418T+
0~0036e-556T+0~002~~e-716T+0~00115e-896r+
. . .J.
By the use of this expression a curve relating v and T can be drawn, froill
which the diffusion constant can he derived for values of v>O.05. (Jf the
expression is to be applied for lower values of ZI account must be taken of
higher terms of the expansion.)
In our experiments the diffusion was measured under conditions as close as
possible to those obtaining in the assay plates. The agar layer was made from
nutrient agar which had been allowed to attain equilibrium with the nutrient
broth used for the overlying liquid layer, by fragmentation of the agar,
followed by 3 days contact a t 2”. In the case of penicillin and dihydrostreptoinycin the concentration in the small samples taken from the liquid layer was
measured by an accurate biological method. In the case of aureomycin the
concentration was measured colorirnetrically, for which purpose concentrations
of the order of 100-500,ug./ml. were required. These concentrations are greater
than can be obtained in nutrient broth and therefore 1 yo KH2P04was used i;
both layers. Control experiments were run to allow for possible inactivation
of the antibiotics during the course of the experiment.
In order to ensure adhesion of the agar to the cylinder wall, the latter was
coated with a layer of ‘ Perspes ’.
EXPERIMENTAL
Theory of di$us ion
All experiments were performed with cups consisting of the ‘fish spiiic ’
insulator beads which are filled by capillarity, and whose effect is to place
a uniform large drop on a uniform small area of the agar surface. Such a bead
is shown in Fig. 1, which also shows the manner in which the contained liquid
enters the agar layer. By the use of isotonic solutions of dyes we found that
under average conditions a bead was empty in 2-3 hr., after which further
spread occurred by simple diffusion. An approximate theoretical treatment
can be derived on the assumption that a known quantity of antibiotic is
placed in the agar a t zero time in the form of an infinitely thin pencil, and
that radial diffusion takes place from the pencil. Although this assumption is
clearly not correct during the first 2 hr., it becomes increasingly true a t later
stages, and permits a t least an approximate prediction of the behaviour of the
diffusing substance.
9-2
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J . H . Humphrey and J . W . Lightbown
We are indebted to Mr A. G. Liddiard of the Department of Theoretical
Physics, King’s College, London, for deriving the following expression which
describes the theoretical behaviour of such a system:
where c = concentration of substance a t a distance r from the centre, Ail = total
quantity of dissolved substance, h =depth of agar layer, D =diffusion constant,
t = time from start.
Initially
After 2-3 hr.
Fig. 1. Diagram illustrating movement of solution from a fish spine bead
into the agar layer.
We found, after this work was completed, that Vesterdal (1947) had used
a similar expression to predict the diffusion of penicillin out of the holes in
a ‘punch’ plate, although he did riot prove its validity in practice. When
Hcatley cups or dry paper disks are used, however, the expression might
require modification.
The expression can be applied to the prediction of the behaviour of inhibition
zones in antibiotic assays on the reasonable assumption that the zone edge
appears where the concentration of antibiotic has a particular critical value
a t a definite time after beginning incubation. This period of time is the ‘ critical
time’, and will in general depend upon the growth characteristics of the test
organism under the conditions of the assay and in certain circumstances upon
the time interval between the appearance of the zone edge and its measurement.
Test of validity of the theoretical expression
The validity of the theory arid the assumption made can be tested by
attempting to deduce from it the zone diameters to be expected for different
antibiotics after allowing various periods of diffusion before incubation. The
thcoretical expression (2)can be rewritten in the form:
r2= 9-21Dt(1og ,?if -log 4nhDtc),
(3)
from which r, the radius of the zone, may be evaluated if the other terms are
known. The values of i%Z and h can be fixed a t will.
The ‘critical ’ concentrations c was determined experimentally by measuring
the minimum inhibitory concentration of antibiotic in pour plates. A layer of
agar medium seeded with the test organism was poured in a series of Petri
dishes. When the agar was set, half was removed from each plate, and the
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Theory of plate assays
133
corresponding spaces were filled by volumes of seeded agar identical in all
respects with that removed, except that they contained graded concentrations
of the antibiotic under test. The plates were incubated under conditions
resembling, as far as possible, those used in our assays, and the growth in the
2 halves was compared. In the case of penicillin and streptomycin a fairly
sharp limiting concentration could be distinguished, above which growth was
almost entirely prevented. With aureomycin, however, although after 8 hr.
incubation a sharp end-point could be distinguished, after 24 hr. incubation
growth had occurred, albeit of diminishing intensity, in a number of higher
concentrations. This was interpreted as being due to instability of aureomycin
in nutrient agar at 35",with consequent loss of its bacteriostatic effect. The
results of our determinations arc given in Table 1.
Table 1. Minimum inhibiting concentrations of antibiotics in agar
plates seedd with test organisms
Penicillin
Streptomycin
Dihydrostreptomycin
Aureomycin HCl
Temp.
38'
28'
38"
38"
35O
B. subtilis I.C.I.
Sarcina lutea
B . subtilis I.C.I.
B. subtilis
Staph. atirciis
Minimum
inhibiting
concentrations
0.018 u./ml.
0.008 u./ml.
0-03u./ml.
0.06 u./ml.
044pg.Iml.
c. 0-10pg./ml.
Time
(hr.)
8-24
16
8-24
8-24
8
24
The diffwsion constanis ( D )were nieasured as described under Methods, and
some results are recorded in Table 2. Most measurements were made a t 4"
because in subsequent experiments diffusion in the cold proved to be necessary
to prevent growth of the test organism and to minimize inactivation of such
antibiotics as are unstable under assay conditions. Diffusion constants are
much affected by changes in the viscosity of water, and hence by changes of
temperature, but corrections for this can be applied on the basis of tables
available in International Critical Tables ( 1929).
Table 2 . Diffusion constants iu 2
Temp.
Penicillin
Penicillin
Penicillin
Aureomycin
Dihydrostreptomycin
20O
37O*
yo agar
Diffusion
constant
( cm.2/hr.)
0.011
0.016
4 O *
0.0066
4 O
0.0075
4 O
0.0053
*
Calculated by comparison with tables of diffusion constants given in International
Critical Tables (1929).
The critical tiwie (i.e. the time aftcr incubation is begun a t which the zone
size and edge are determined) was not easy to determine in practice. The
problem was approached in two ways. In the first, which was used by Cooper
& Woodman (1946), assay plates were poured containing seeded agar, and
beads containing a constant amount of antibiotic solution were placed on them
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134
J . H . Humphrey and J . W. Lightbown
at varying intervals of time after incubation was begun. It was argued that
beads placed after the critical time should show no inhibition zones. The
drawback to this method lay in deciding whether to take account of small
zones of inhibition which appeared even after general growth had occurred,
and were associated with such high antibiotic concentrations that they were
probably due to lysis. The second method consisted in placing many beads
containing a constant concentration of antibiotic upon an assay plate. The
plate was incubated at 37", and from it a t varying intervals of time, beads were
removed, together with the agar immediately below and around them, by
means of a cork borer. After the critical time removal of the bead may be
expected not to affect the zone size. Estimates of the critical time obtained
by the second method have always been about 1 hr. shorter than those by the
first method, and for practical purposes we have taken a mean value. Since
the concept of a sharp critical time is somewhat abstract, it is doubtful
whether greater precision could be obtained.
With the values obtained it was possible to predict theoretically the zone
diameters when antibiotics were allowed to diffuse for varying periods of time
into seeded agar plates in the cold (i.e. without growth of the test organism)
before incubation was begun. In order to permit sufficient replicate beads for
accurate measurement a t each time interval, it was necessary to use two assay
plates in a single experiment. These were prepared as alike as possible, and one
of the sets of replicate beads was repeated on each plate, as a check to ensure
satisfactory duplication. In Fig. 2 are shown the experimental zone diameters
observed a t two levels of aureomycin concentration in relation to those
predicted theoretically using the value for the diffusion constant obtained
a t 4". Although failure to correct for the increased diffusion constant during
the incubation period entails some error in the theoretical curves, it will not
be great since the incubation period was short compared with the total period
allowed for diffusion. It will be seen that the experimental points lie fairly
close to the predicted curves, although a better fit is obtained with a slightly
lower value for the critical concentration than that recorded in Table 1.
Fig. 3 shows a similar experiment, using penicillin with B. subtilis as test
organism, conducted throughout at 37". Because growth occurred throughout
the experiment at this temperature the observation period was limited to
3$ hr. By using short time intervals and many replicate beads we obtained
enough experimental points to show that under these conditions also agreement with the theory was remarkably good, except when the period of diffusion
was brief. Since in this part of the curve the time interval between placing
beads on the agar and the appearance of inhibition zones was too short to allow
complete entry of the fluid into the agar, the conditions assumed in the
theory did not apply, and the aberrant points can be discounted.
Consequences of the theory
Regarding the theory as substantially valid we can deduce some practical
consequences which will apply not only to inhibition zones produced by
antibiotics but also to exhibition zones produced by growth factors.
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Theory of plate assays
135
(i) Fig. 4 shows some distribution curves of antibiotic in the agar around
a bead, calculated for varying intervals of time. The figure in question relates
to penicillin, but the general form of the curves applies to other substances,
and it emphasizes how greatly the concentration a t the centre exceeds that at
the boundary. In the case of the diffusion assay of growth factors it is clear
that there is little likelihood of organisms which grow inside the zone exhausting
the growth factor to any significant extent.
(ii) From equation (3) it is apparent that under any given set of assay
conditions ~2 should be proportional to log M.
40
-
30 h
€
E
v
L
0
;
20m
U
.-
D
al
5
N
10
-
0
10
350
I
6
>:
I
0
I
10
40
I
20
I
30 hr.
Fig. 2. Zone diameters observed after varying periods of diffusion in the cold followed by
incubation at 35", compared with diameters predicted theoretically. Antibiotic :
aureomycin; test organism: Staph. aureus. x x and 00 are experimental points for
beads containing solutions of 25 and 6%5pg./ml. respectively. Each point is the mean
of 5 diameters, which were read after 8 hr. incubation. The theoretical curves are drawn
from the expression
r2=9*21Dt(lOgM - log 41~hDi!Q),
where D=0.0075 cm.%/hr.,h = 0 4 cm., and iM=0.625 and 0-156pug. --, curves
when a=O.O25pg./ml.; ----, curves when (T = O-O%pg./rnl.
There has been much discussion as to whether the relationship between the
log. of the antibiotic concentration and zone diameter is linear or whether
a square term is involved. The general, although not the universally agreed,
conclusion is that in penicillin assays the relationship is linear. It must be
borne in mind, however, that a significant departure from a linear relationship
will be noticed only if an assay is very accurate (in the sense that the coefficient
of variation of a single zone is small and many replicates are used) or if a wide
range of concentrations of antibiotic is used. We have found that when very
accurate assays were performed (using 64 replicates a t each dose level) the
logarithm of the antibiotic concentration was more nearly proportional to the
square of the zone diameter than to the zone diameter in the following assays :
streptomycin and dihydrostreptomycin with B. subtilis as test organism;
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J . H . Humphrey and J . W . Lightbown
136
aureomycin and terramycin with B. pumilus; penicillin with Staph. aureus.
When penicillin was assayed with B. subtilis I.C.I. strain the zone diameter,
and not its square, gave the best fit. A possible reason for this discrepancy,
which may have wider application, is given below.
I
0
I
1
I
I
l
l
I
2 3 4 5 6
Diffusion time (hr.)
l
7
L
8
Distance from centre (mm.)
Fig. 3
Fig. 4
Fig. 3. Zone diameters observed after varying periods of diffusion at 37",compared with
those predicted theoretically. Antibiotic : penicillin ; test organism : B. subtilis. Each
experimental point is the mean of 16 beads, containing a solution 16 u./ml. The
theoretical curve was plotted from
ra=9*2lD)t(logM-log 4nhDta),
where D=0.016 cm.a/hr., M=0-4 unit, h=0.4 cm., a=0.02 u./ml.
Fig. 4. Curves illustrating the distribution of penicillin concentration in an agar layer depth
0.4 cm. around fish spine beads containing penicillin solution, after 3, 7 and 20 hr.
diffusion at room temperature. -, bead with 16 u./ml.; - - - -, bead with 4 u./ml.
The critical concentration is 0-02u./ml.
(iii) The sharpness of the edge of a zone, apart from effects due to lysis, will
depend upon the range of antibiotic concentrations over which partial but not
complete inhibition of growth occurs. Suppose that u2is the critical concentration which permits no growth, and clthe concentration which permits full
growth of the test organism. Then over the range ul to u2a shading of growth
will occur. Let rl and r2 be the corresponding distances from the cup, so that
the width of the edge is r1-r2. From the expression (3)it can be shown that
(z),
r p 2= 9-21Dt
log
212
where K is the mean of rl and r 2 . If therefore the response of the organism to
the antibiotic cannot be made homogeneous (e.g. by single colony selection),
the effects on the zone edge will be minimized when the diffusion time is short
and the zone diameter large.
~
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Theory of plate assays
137
(iv) Factors governing the slope of the dosage-response curve can be
evaluated as follows. If d , and d2 are the zone diameters when the amounts of
antibiotic in the cup are MI and M , , it is readily shown that
d,-d,=--- 9.21 Dt (log MI-log M , ) ,
R
(4)
i.e.
-
10-
E
E
;8 W
0
W
5
6-
c
2
c
F
4-
Y
u
n
u
g
en
2-
0~""""""'"'
Diffusion time (hr.)
Fig. 5. Theoretical curves demonstrating the relationship between slope and diffusion time
for a penicillin assay at levels of 16 and 4 u./ml. Curves are shown for three different
temperatures of diffusion. No correction of the 4 and 20' curves is made for subsequent
incubation at 37'. The values used for D are given in Table 2.
From (4) it appears that the slope will vary directly with diffusion tinie, and
inversely with the mean zone radius. It will also depend directly upon the
diffusion constant. Two practical conclusions can be drawn. The first is that
when assays of several antibiotics are preformed under standard conditions,
and when, as occurs in practice, zone diamcters are chosen for convenience to
be about the same, the main factor governing the slope will be the diffusion
constant. The commoner antibiotics all have similar diffusion constants, and
the slopes in their assays will consequently also be similar.
The second conclusion is that the slope can be increased by prolonging the
diffusion time and by keeping the antibiotic concentrations and therefore the
zone diameters small. In practice, the value of such measures can be limited
by loss of definition of the zone edge, for they are precisely opposite to the
recommendations of 0 (iii). Fig. 5 , however, illustrates the effect upon the
slope of penicillin assays of the length of the diffusion time a t different
temperatures.
The special cases of penicillin and streptomycin assays
When penicillin assays are performed with the I.C.I. strain of B. subtilis,
and streptomycin assays with B. subtilis or other Gram-positive organisms,
the zone edges are characteristically very sharp; they do not shift with
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J . H . Humphrey and J . W. Lightbown
continued incubation; and contiguous zones do not distort one another. These
observations suggested that there might be a sharp drop in antibiotic concentration at the zone edge, which might be due either to its destruction or to its
removal by the organisms at that site. Investigations have shown such a drop
to occur, although the mechanisms in the two cases are different and must
be treated separately.
Penicillin assays
An experiment was devised to discover whether penicillin did in fact diffuse
beyond the zone edge of B. subtilis seeded plates. For this purpose plates were
prepared which were seeded with a mixture of I.C.I. strain B. subtilis spores
and of S . Zutea. The sarcina is about three times as sensitive as the subtilis to
penicillin, and the two organisms grow well in each other's presence. This
Incubated at 28"
c
Incubated at 3 5 '
C
Fig. 6. Scale diagram of an experiment which illustrates the failure of penicillin t o diffuse
past zone edges formed by B. subtilis. Matched plates were poured with agar seeded
with B. subtiZ,isalone ( A ) ,Sarcina Zutea alone ( B ) ,and a mixture of the two (C).Beads
containing 16 u./ml. penicillin were placed on each plate and the inhibition zones were
measured at intervals during incubation at 28 or 35O. Sarcina alone gave a wide zone
which increased with time owing t o lysis. I n the mixed plate a t 28O the Sarcina zone
appeared first, but became fixed when the B. subtilis zone appeared inside it. A t 35"
growth of both organisms became apparent a t the same time, and Sarcina appeared and
remained present right up t o the B. subtilis zone edge.
point was verified by examination of stained smears from the agar. Beads
containing a fixed quantity of penicillin were placed upon plates seeded with
the mixture of organisms, and upon control plates containing each of the
organisms alone but in other respects identical. Sets of plates were incubated
at 28 and 35", since by varying the temperature it was possible to accelerate
or retard the growth of B. subtilis relative to that of S . lutea. The plates were
examined a t intervals, and Fig. 6 illustrates diagrammatically the behaviour
of the inhibition zones after 7 and 24 hr. The S . Zutea zones shifted outwards as
incubation continued, whereas the B. subtilis zones remained fixed. When both
organisms grew together the S . lutea zones became fixed, and the size a t which
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Theory of plate assays
139
they were fixed depended upon the time at which significant growth of
B. subtilis had occurred. The most probable explanation is that penicillin did
not get past the B. subtilis zone edge.
Reinoval of penicillin by adsorption by the organisms might provide
a possible explanation of such an effect, but calculations based on the data
of Rowley, Cooper, Roberts & Smith (1950) showed that the amount adsorbed
would be far too small. We were therefore led to test for penicillinase production. In order that conditions of growth of the B. subtilis should be comparable
to those obtaining in assays, plates seeded with spores were incubated under
standard conditions. After 16 hr. the agar was removed and frozen solid in
a closed container, after which it was allowed to thaw. The liquid formed was
collected, and was sterilized by filtration through a Gradocol membrane.
Aliquots of the liquid were incubated a t 37" in the presence of phosphate
buffer p H 7.0 for varying periods of time with varying amounts of penicillin.
The amount of penicillin destroyed was estimated biologically. In a typicaI
experiment with a filtrate from the I.C.I. strain the rate of inactivation of
penicillin. present at an initial concentration of 16 u./ml., was 28 u./ml.
filtrate per hour. The enzymic nature of this inactivation was confirmed by
showing that the extent was proportional to the time of incubation. Such
a low penicillinase activity would escape detection by most methods, but it is
sufficient. nevertheless, to destroy penicillin at the zone boundary, as the
following calculations show.
Consider an annular ring of agar 25 mm. diam., of depth 4 mm. and of
width 0.051nini. This is approximately the size of the boundary area of a typical
inhibition zone, and its volume is 0.015 C.C. When once the organisms have
grown, this volume of agar will be able to inactivate approximately 0.42 11.
penicillin/hr. Calculation of the amounts of penicillin which would enter such
a ring from a bead a t the centre containing a solution of penicillin 16 u./ml.
showed them to be as follows: after 3 hr., nil; after 7 hr., 0.0174 u.; after
11 hr., 0.06'7 ti. ; after 20 hr., 0.19 11. Even if the penicillinase were considerably
less active a t lower concentrations of penicillin, it would be adequate to
inactivate such small amounts.
Although the above considerations suggest that penicillinase production by
the test organism will result in sharp zone edges it must be emphasized that
only minimal penicillinase production is likely to produce this effect. The
Mill Hill strain of B. purnilus can be shown to produce more potent or more
diffusible penicillinase than the I.C.I. strain, and when it is used the zone
edges are somewhat ragged because of small colonies which grow up inside the
main zone boundary. Ingrani (1951) illustrates well the appearance of
scattered colonies inside the inhibition zone with a penicillinase-producing
resistant staphylococcus, and remarks that fine crenation of the edge niakes
i t unsuitable for assay work.
Streptomycin assays
We were unable to demonstrate enzymic inactivation of streptomycin or of
dihydrostreptomycin by any of the test organisms. It was known, however,
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J . H . Humphrey and J . W . Lightbown
from the work of Linz (1949) that Gram-positive organisms adsorb streptomycin, and we therefore studied this adsorption quantitatively. Thick suspensions were prepared of B. subtilis, B. pumilus and of Staph. aureus grown
for 16-20 hr. on slopes of assay agar. The suspensions were washed with water,
and dried from the frozen state. Varying amounts of the organisms were
incubated with varying concentrations of streptomycin in phosphate buRer
p H 7.8 for periods ranging from 10 min. to several hours. The organisms were
21
- 0
0
'
I
1
log (concentration of free streptomycin in u.lm1.)
I
2
Fig. 7. Relationship between streptomycin adsorbed per mg. dry weight of organisms and
concentration of free streptomycin in the medium. Experiments conducted in phosphate buffer pH 7.8, ,u=O-1 at 37". The results are plotted on a logarithmic scale:
0-0,Staph. aureus; x -x , B. subtilis.
removed by centrifugation and the concentration of streptomycin remaining
in the supernatant was estimated biologically (fiducial limits of the assay not
more than f3 for P -c0.05). It was found that removal of streptomycin by
the organisms was independent of time of contact, but that the amount
removed per unit weight of organisms depended upon the concentration of
streptomycin present. Fig. 7 illustrates the results of experiments with Staph.
aureus and B. subtilis. The units of' streptomycin adsorbed per mg. dry weight
of organisms are plotted on a double logarithmic scale against the concentration of streptomycin in u./ml. remaining in the supernatant fluid. It will be
seen that in both cases the adsorption fits an adsorption isotherm of the
Freundlich type. The equations describing their behaviour are : x =20.4 CoM
for Staph. aureus and z = 1.82 CO'68for B. subtilis, where x =units adsorbed/mg.
and C=u./ml. in the supernatant. The greater adsorption by Staph. aureus is
probably associated with the fact that the suspension used contained much
more Gram-positive material than the B. subtilis.
Adsorption of streptomycin in this way could account for a sharp fall in
concentration a t the zone edge, as the following illustrative calculation shows.
Let c be the concentration which would obtain a t a given distance from
a bead in the absence of adsorption by the organisms, and let C be the actual
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Theory of plate assays
141
concentration of free streptomycin in the presence of organisms. (c- C ) will
be the amount of streptomycin adsorbed by the organisms present in 1 ml.
of the agar layer.
The organisms present in 1 ml. after a time t hr. will be q.2(t-*1)'g,where
4 =initial inoculum in mg./ml., t * = lag period, g = mean generation time. From
the adsorption isotherm we have
where
Q
and b are constants. We also have, froni the diffusion equation,
Hencc
From this equation, using the values obtained experimentally, it is possible to
calculate the true concentration C' at a distance r froni the bead after a given
time t .
Consider a bead containing Streptomycin 80 u./ml. placed upon a cold seeded
agar plate which is then incubated a t 37". We know from experience t h a t the
zone diameter will be about 2.15 cm. This diameter is, incidentally, almost
equal to the calculated value for D = 0 - 0 1 2 cm.2/hr., critical time 45 hr., and
critical concentration 0.03 u./ml. Let us considcr then the change of actual
concentration with time at a distance r = 1-07 cni. from the bead, and substitute
the following values in the equation : Jl = 2 units (nieasured), h = 0.4 cni.
(measured). D =0.012cni.2/hr. (estimate based on measurement a t 4O),
q = 10-4 mg. (measured), t 1 = 1 hr. (estiniated), g = 0-4 hr. (nieasured in shaken
broth culture), a = 1.82 (measured), b = 0.68 (measured).
Calculations give the following results :
C = actual
concentration
(u./ml.)
r = uncorrected
concentration
(u./nil,)
0.00000000.5
0*00cH)0000;3
0.00013
0-0041
0.018
0.037
0.02'2
0.001:3
04048
0.02
0.06
0.11
After 6&hr., which is equivalent to 13 generations, the density of the organisms
in the surface layer would be approximately 2 mg./ml., which is of the order
of the maximal growth to be expected, and i t is evident t h a t division would
cease, and the equation would no longer be applicable. The calculations entail
too many approximations to permit exact conclusions, but they show, nevertheless, t h a t once growth begins, at the site which is later t o contain the
visible zone edge, the adsorption of streptomycin by the organisms already
present can, while growth continues, keep the concentration of free streptomycin from significantly exceeding the critical concentration (0.03 u./ml.).
I n the case of Staph. aureus, which adsorbs streptomycin more powerfully,
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142
J . H . Humphrey and J . W . Lighfbown
the same argument can be used but the more powerful adsorption will permit
the zone diameter to be much smaller-as in practice it is. We have observed,
furthermore, that when streptomycin is added to a plate seeded with B. subtilis
at 38" after growth of the test organisms has taken place, no macroscopic lysis
occurs. Hence it is immaterial, once the visible zone edge is established, how
far streptomycin later penetrates beyond it.
DISCUSSION
To persons experienced in assay procedures inany of the conclusions which
have been reached above from theoretical reasoning may appear to be matters
of common sense. There is some advantage, however, in being able to consider
quantitatively some of the factors governing zone diameters. For example it
is possible from a knowledge of the zone diameters produced by two different
concentrations of a substance, and of the time of diffusion involved, to calculate
its diffusion constant. For obvious technical reasons the value obtained in this
way may not be very accurate, but it will be obtained with little trouble.
It is also an advantage to be aware of the factors concerned in determining the
zone edge, since it becomes clear that the ideal of a perfectly sharp zone edge
requires either a homogeneous population of test organisms with a sharply
defined critical concentration and no tendency to lysis, or the existence of
special circumstances such as we have demonstrated in the case of our penicillin
and streptomycin assays. In the absence of such circumstances, although of
course such devices as layered plates give some improvement, attempts to
secure perfect zone edges may lead only to frustration. In the case of aureomycin assays, for example, we have been unable to demonstrate either
' aareomycinase ' activity, or adsorption of aureomycin, by our test organisms,
a.nd have had to be content with attempts to improve the zone edge by other
methods such as the use of a test organism which after growth produces an
alkaline reaction, and thereby swings the pH in a direction away from the
optimum for aureomycin (Valentine & Johns, 1949).
The prediction that the log. dose will be proportional to the square of the
zone diameter, and its experimental verification in most of our assays, may
appear to be contrary to common experience, particularly since it should apply
to punch plate assays also. This expected relationship would of course become
invalid if any factor were to affect small and large zones unequally. As may
be seen from Fig. 4,after a given period of diffusion the rate of increase of antibiotic concentration within the zone boundary is greater for the larger amount
of antibiotic than for the smaller. The effect of production of a small excess of
penicillinase a t the zone edge, capable of diffusing inwards, would therefore
be to decrease the smaller zones more than the large. Although we have not
been able to treat this effect mathematically, it is in the direction which would
be required to produce alinear relationship between log. dose and zone diameter,
which is that commonly assumed and which we also observed.
From a practical point of view, the distinction between a linear and
a quadratic relationship between the log. dose and the zone diameter may
prove to be unimportant except when great accuracy is sought. As an example
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Theory of plute assays
143
let us consider a n assay in which the mean zone diameter for a given dose is
2-00 em. and for 4 x the dose 2-50 cm. If the log. dose is proportional to the
diameter, the expected mean diameter for 2 x the dose will be 2.25 cm.,
whereas if the log. dose is proportional to the square of the diameter the expected
mean diameter for 2 x the dose will be 2.264 cm. In order to distinguish
between these two values a variation of 0.014 cni. in the mean diameter for
2 x the dose must be significant-i.e. the coefficient of variation must be less
than 0.25 yo.This condition would require that a t least16 replicates be used at
each dose level if a single plate were used, and more if several plates were
involved.
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D. (1946). The diffusion of antiseptics through agar
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FRIEDMAN,
L. & KRAEMER,
E. 0. (1930). The structure of gelatin gels from studies
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0. (1951). An antibiotic assay tray. J . gen. Microbiol. 5 , 357.
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R. (1949). Sur le mecanisme de l’action de la streptomycine. 1. Action de la
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H.W. & WEAVER,W. (1928). The diffusion problem for a solid in contact
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ROWLEY,D., COOPER,
P. D., ROBERTS,
P. W. & SMITH,E. L. (1950). The site of
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R. G. S. (1949). A suggested method for the titration
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(Received 16 February 1952)
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